U-shaped flow profile

Liquid plug flow produces a horizontal (i.e., flat) flow profile (Fig. 7.7a), Channeling produces a U-shaped flow profile (Fig, 7.8a). The liquid moves fast at the tray center and slow near the walls. Wide stagnant zones, a steep U shape, and liquid recirculation in the stagnant zones (Fig. 7.35) signify a highly channeled flow profile. The... 

Shape factors of a different sort are involved in the Taylor dispersion problem. With parabolic flow at mean speed U through a cylindrical tube of radius R, Taylor found that the longitudinal dispersion of a solute from the interaction of the flow distribution and transverse diffusion was R2U2/48D. The number 48 depends on both the geometry of the cross-section and the flow profile. If, however, we insist that the flow should be laminar, then the geometry of the cross-section determines the flow and hence the numerical constant in the Taylor dispersion coefficient. 

Fig. 5. Elution profiles obtained for the tryptic digest peptides of hemoglobin A using three distinct gradient shapes (number 3,6, and 8 on the Waters M660 solvent programmer). In each case, two /u-Bondapak C, columns were connected in series, a flow of 1.7 ml/min was used, and a 30-min linear gradient was generated from solvent A (0.phosphoric acid, pH 2.2) and solvent B (SOOt acetonitrile-0.1% phosphoric acid, pH 2.2). Reprinted with permission from Bishop et al. (5. ). Copyright by Marcel Dekker, Inc., New York.

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

For a first-order irreversible reaction in a packed bed where axial dispersion is significant, the concentration profile has the shape shown in Figure 6.11. The material balance for a differential element is given in Eq. (6.19), where D is the same as the effective axial dispersion coefficient, and u is the superficial velocity. The temperature, pressure, and molar flow rate are assumed not to change, so u and k are constant, and the equation is written for a unit cross section of the reactor ... 

The variables. A, S, x, and u, are the flow cross section, the wetted perimeter, the shear stress, and the average velocity of the two fluids x. is the interfacial shear stress where a positive z. corresponds to a faster upper layer. The shape factors, y, Y are defined in terms of local velocity profiles, u ... 

Figure 10 shows the measured dimensionless velocity profiles (U/Umax) for the free flow velocity range of 23-177 m/s. Figure 11 shows the measured dimensionless RMS turbulent velocity profiles (u /u max) for the corresponding free flow velocities. Experimental results demonstrate that all of the profiles of flow velocity have a similar shape and that all of the profiles of RMS turbulent velocity have a similar shape. 

In Eq. 5.10a, the factor f /3 j corresponds to a Poiseuille flow, with no slip at the solid. Therefore, the validity of Eq. 5.10a is restricted to mesoscopic films. However, for molecular ones, the proportionality between D(f) and dU/d is retained. Note that as the droplet shape f(x, t) evolves slowly in time, U(f) 9f/9x evolves slowly in space therefore the profile f (x, t) gives qualitative information on the shape of the disjoining pressure ellipsometric droplets profiles are not merely striking pictures. Let us comment further on short-range contributions. 

In the equation, k=k rj denotes the permeability of the porous media, t] the fluid viscosity, p is the pressure and u represents the superficial flow velocity of the fluid. The permeability, k, taken as a constant, is an intrinsic property of the porous medium. The velocity in Darcy s law, m, is actually a mean velocity over the cross-section. Hence, the velocity profile of Darcy flow in a porous medirnn has a flat shape in the flow direction, indicating a constant velocity. 

The resistance to flow is the property of a sealant to remain in the specified shape after processing. For testing, a U-profile is filled with sealant (EN 27390, DIN 52454, ISO 7390, ASTM D-2202). 

---